Intersection problem for Droms RAAGs

TL;DR

This paper presents an algorithmic solution to the subgroup intersection problem for Droms RAAGs, enabling the determination and computation of intersections of subgroups within these groups.

Contribution

It introduces a method to decide and compute subgroup intersections for Droms RAAGs, extending understanding of their algebraic properties.

Findings

Decidable subgroup intersection problem for Droms RAAGs

Algorithm computes generators for subgroup intersections when finitely generated

Solvability of SIP passes through free and direct products with free-abelian groups

Abstract

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether $H \cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H \cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable SIP.